Without the loss of generality, it is conventional for the
translational invariant system to impose the periodic boundary
condition. In spite of the simplicity, the Hamiltonian Eq.
(REF ) has a wide range of applications, such as the Bogoliubov-
de Gennes Hamiltonian in superconductivity, graphite systems
[1]}. Applying the Fourier transformation and considering the
periodic boundary condition, \(\mathbf {c}_x=1/\sqrt{N}\sum _ke^{ikx}\mathbf {c}_k\)  in which \(k=2\pi n/N\)  with \(n=1,2,\dots , N\) .
Then the Hamiltonian in the momentum space can be written generally
as \(H = \sum _k \mathbf {c}_k^{\dagger }{H}(k)\mathbf {c}_k\) . If
one introduces the four-vector \(R_{\mu }(k)(\mu =0,x,y,z)\) , then the
Hamiltonian can be rewritten as,
\(H=\sum _{\mu }\sum _k\mathbf {c}_k^{\dagger }R_{\mu }(k)\sigma _{\mu }\mathbf {c}_k,\) 
